Optimal. Leaf size=286 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.134835, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{x^{7/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{a b+b^2 x} \, dx}{9 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{a b+b^2 x} \, dx}{9 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^2 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{9 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a^3 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{9 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^4 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{9 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (4 a^4 \left (\frac{9 A b^2}{2}-\frac{9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{9 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x} (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.083478, size = 139, normalized size = 0.49 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (21 a^2 b^2 x (5 A+3 B x)-105 a^3 b (3 A+B x)+315 a^4 B-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )-315 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{315 b^{11/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 197, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{315\,{b}^{5}} \left ( 35\,B\sqrt{ab}{x}^{9/2}{b}^{4}+45\,A\sqrt{ab}{x}^{7/2}{b}^{4}-45\,B\sqrt{ab}{x}^{7/2}a{b}^{3}-63\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+63\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+105\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-105\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-315\,A\sqrt{ab}\sqrt{x}{a}^{3}b+315\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b+315\,B\sqrt{ab}\sqrt{x}{a}^{4}-315\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36375, size = 617, normalized size = 2.16 \begin{align*} \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20385, size = 277, normalized size = 0.97 \begin{align*} -\frac{2 \,{\left (B a^{5} \mathrm{sgn}\left (b x + a\right ) - A a^{4} b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) - 45 \, B a b^{7} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + 45 \, A b^{8} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + 63 \, B a^{2} b^{6} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) - 63 \, A a b^{7} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{5} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{6} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 315 \, B a^{4} b^{4} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - 315 \, A a^{3} b^{5} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{315 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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